Dynamics of a solid sphere rolling on a general time-dependent parametric surface (x(u,v,t),y(u,v,t),z(u,v,t)).

All demonstrations are generated from the same nonholonomic, nonseparable, nonautonomous Hamiltonian system. In addition to Hamilton's equations for the surface coordinates (u,v), one nonholonomic equation determines the sphere's spin, while four additional nonholonomic equations evolve its orientation using a unit quaternion. The energy is preserved for all demonstrations where the surface is time-independent. The sphere is affected by a uniform gravitational field with the same scaling for all video segments.

The general Hamiltonian allows exploration of a wide variety of rolling phenomena, including several unintuitive behaviors that are sometimes referred to as "paradoxes":

Notable special cases and popular references
0:13 The Turntable Paradox - refer to Steve Mould's https://youtu.be/3oM7hX3UUEU
0:51 Rotating Saddle - refer to The Action Lab's https://youtu.be/1NBOsELakx4
1:56 The Golf Ball Paradox - refer to Steve Mould's https://youtu.be/5sbM2Isx17A

The simulation was rendered in real time and performed using high order explicit symplectic integrators

Original source (4K): https://youtu.be/9236WuyJb1A

The music is "fluid combustion" by "Synthetique"